Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489691 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #33771 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
776 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(18,23)(19,22)(20,27)(24,32)(25,31)(26,29)(28,30);(4,13)(8,12)(9,16)(11,14)(17,18)(19,24)(20,28)(21,25)(22,30)(23,27)(26,31)(29,32);(1,3,5,10)(2,6,7,15)(4,14,13,11)(8,16,12,9)(17,19,21,22)(18,24,25,30)(20,26,29,27)(23,28,31,32);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^9|c^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72321431 |
| maximal normal subgroup of | #72321431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397009);